Simplify and expand the following expression: $ \dfrac{z}{2z + 10}-\dfrac{z}{5z + 8} $
Explanation: In order to subtract expressions, they must have a common denominator. Get both fractions over a common denominator of $(2z + 10)(5z + 8)$ Multiply the first term by $\dfrac{5z + 8}{5z + 8}$ $ \begin{align*} \dfrac{z}{2z + 10} \times \dfrac{5z + 8}{5z + 8} & = \dfrac{(z)(5z + 8)}{(2z + 10)(5z + 8)} \\ & = \dfrac{5z^2 + 8z}{(2z + 10)(5z + 8)}\end{align*} $ Multiply the second term by $\dfrac{2z + 10}{2z + 10}$ $ \begin{align*} \dfrac{z}{5z + 8} \times \dfrac{2z + 10}{2z + 10} & = \dfrac{(z)(2z + 10)}{(5z + 8)(2z + 10)} \\ & = \dfrac{2z^2 + 10z}{(5z + 8)(2z + 10)}\end{align*} $ Now we have: $ = \dfrac{5z^2 + 8z}{(2z + 10)(5z + 8)} - \dfrac{2z^2 + 10z}{(5z + 8)(2z + 10)} $ Now both terms have a common denominator we can subtract the numerators: $ = \dfrac{5z^2 + 8z - (2z^2 + 10z)}{(2z + 10)(5z + 8)} $ $ = \dfrac{5z^2 + 8z - 2z^2 - 10z}{(2z + 10)(5z + 8)} $ $ = \dfrac{3z^2 - 2z}{(2z + 10)(5z + 8)}$ Expand the denominator: $ = \dfrac{3z^2 - 2z}{10z^2 + 66z + 80}$